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A property $\mathcal{P}=({\mathcal{P}}_n)$ of $n$-variable Boolean functions is:

A property satisfying all three conditions is a natural proof. The Razborov–Rudich natural-proofs barrier states that if exponentially secure pseudorandom function families exist in $\mathrm{P}/\mathrm{poly}$, then no property that is constructive and large is useful against polynomial-size circuits.

Suppose such a property $\mathcal{P}$ existed. Given oracle access to an unknown $n$-variable function, query its full truth table and run the constructive membership test. This takes $2^{O(n)}$ time. For a truly random function, largeness makes the test accept with probability at least $2^{-O(n)}$. Repetition amplifies this to a constant acceptance probability within $2^{O(n)}$ time.

For a function drawn from the assumed pseudorandom family, each keyed function has polynomial-size circuits, so usefulness makes the test reject for the relevant lengths. The amplified membership test therefore distinguishes the pseudorandom family from a truly random function with constant advantage in exponential time, contradicting exponential pseudorandomness. Hence the three conditions cannot coexist.

Steven Rudich coauthored the natural proof framework with Alexander A. Razborov. Their 1997 paper identifies a barrier to a broad form of circuit lower bound argument, conditional on a sufficiently strong pseudorandom function family security assumption.^[ref]

A property that is efficiently recognizable from a truth table, holds for many random Boolean functions and excludes the target small-circuit class could distinguish pseudorandom function families from random ones. The required security must cover the time spent reading the entire table, which is exponential in the function's input length. The barrier restricts these proof methods under its assumption; it does not prove that circuit lower bounds are impossible.